Existence of non-constant curvature metrics on RP^2 with ω1=ω2=ω3 equal to twice the systole

Determine whether there exist Riemannian metrics g on the real projective plane RP^2 such that the first three widths ω1(RP^2,g), ω2(RP^2,g), and ω3(RP^2,g) all equal 2·sys(RP^2,g), where sys(RP^2,g) denotes the length of the shortest non-contractible loop, and yet the metric g does not have constant sectional curvature.

Background

In Section 4, the authors analyze the implications of having the first three widths equal and employ min-max and geometric arguments to obtain rigidity under certain conditions. They show that when the entire volume spectrum matches that of the canonical RP^2, the metric must be isometric to the standard constant curvature one.

They then highlight a more specific scenario relating widths to the systole and explicitly note that they do not know of any examples where ω1=ω2=ω3 equals twice the systole without constant curvature, leaving open the existence of such metrics.